Wick product
Encyclopedia
In probability theory
, the Wick product
named after physicist Gian-Carlo Wick
, is a sort of product
of the random variable
s, X1, ..., Xk, defined recursively as follows:
(i.e. the empty product
—the product of no random variables at all—is 1). Thereafter we must assume finite moments
. Next we have
where
means Xi is absent, and the constraint that
and the angle-bracket notation
is used to denote the expected value
of the random variable X.
with n factors.
The sequence of polynomials Pn such that
form an Appell sequence, i.e. they satisfy the identity
for n = 0, 1, 2, ... and P0(x) is a nonzero constant.
For example, it can be shown that if X is uniformly distributed
on the interval [0, 1], then
where Bn is the nth-degree Bernoulli polynomial. Similarly, if X is normally distributed with variance 1, then
where Hn is the nth Hermite polynomial.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, the Wick product
named after physicist Gian-Carlo Wick
Gian-Carlo Wick
Gian Carlo Wick was an Italian theoretical physicist who made important contributions to quantum field theory...
, is a sort of product
Product (mathematics)
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...
of the random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s, X1, ..., Xk, defined recursively as follows:
(i.e. the empty product
Empty product
In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is equal to the multiplicative identity 1, given that it exists for the multiplication operation in question, just as the empty sum—the result of adding no numbers—is zero, or the additive...
—the product of no random variables at all—is 1). Thereafter we must assume finite moments
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...
. Next we have
where
means Xi is absent, and the constraint thatExamples
It follows thatAnother notational convention
In the notation conventional among physicists, the Wick product is often denoted thus:and the angle-bracket notation
is used to denote the expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of the random variable X.
Wick powers
The nth Wick power of a random variable X is the Wick productwith n factors.
The sequence of polynomials Pn such that
form an Appell sequence, i.e. they satisfy the identity
for n = 0, 1, 2, ... and P0(x) is a nonzero constant.
For example, it can be shown that if X is uniformly distributed
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...
on the interval [0, 1], then
where Bn is the nth-degree Bernoulli polynomial. Similarly, if X is normally distributed with variance 1, then
where Hn is the nth Hermite polynomial.
Binomial theorem
Wick exponential